Question 1101530: 2logx+8=log4
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! this can be interpreted as:
2 * log(x) + 8 = log(4)
if so, then:
subtract 8 from both sides of the equation to get:
2 * log(x) = log(4) - 8
since 2 * log(x) is equal to log(x^2), then:
log(x^2) = log(4) - 8
by the basic definition of logs, this is true, if and only if:
10^(log(4)-8) = x^2
solve for x to get:
x = plus or minus square root of (10^(log(4)-8)
solve for x to get:
x = plus or minus .0002
x can't be negative because log(x) can't be negative, therefore x = .0002.
confirm by replacing x with .0002 in the original equation.
2 * log(.0002) + 8 = .6020599913
log(4) = the same.
solution is confirmed.
the problem could also be interpreted as:
2 * log(x+8) = log(4)
in that case:
divide both sides of the equation by 2 to get log(x+8) = log(4)/2
this is true if and only if 10^(log(4)/2) = x+8
solve for x to get x = 10^(log(4)/2)-8 = -6
you get x = -6
replace x in the original equation of 2 * log(x+8) = log(4) with -6 to get:
2*log(-6+8) = .6020599913
log(4) = .6020599913
they're the same so the solution is confirmed to be good.
it is always good to use parentheses to ensure that what you are asking is interpreted as such by the people you are asking it to.
you have two possible answers, depending on what your original equation was asking.
2 * log(x) + 8 = log(4) yields x = .0002
2 * log(x+8) = log(4) yields x = -6
the general logarithmic equation properties used are:
log(x^n) = n*log(x)
log(x) = y if and only if x = 10^y
the more general form of the last statement is:
logb(x) = y if and only if x = b^y
when you use the form of log(x) without mentioning the base, the base is assumed to be 10.
the log function of your calculator assumes the base is 10.
log(x) means the same thing as log10(x).
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